Optimal non-symmetric Fokker-Planck equation for the convergence to a given equilibrium

Abstract

<p style='text-indent:20px;'>This paper is concerned with finding Fokker-Planck equations in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^d $\end{document}</tex-math></inline-formula> with the fastest exponential decay towards a given equilibrium. For a prescribed, anisotropic Gaussian we determine a non-symmetric Fokker-Planck equation with linear drift that shows the highest exponential decay rate for the convergence of its solutions towards equilibrium. At the same time it has to allow for a decay estimate with a multiplicative constant arbitrarily close to its infimum.</p><p style='text-indent:20px;'>Such an "optimal" Fokker-Planck equation is constructed explicitly with a diffusion matrix of rank one, hence being hypocoercive. In an <inline-formula><tex-math id="M2">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>–analysis, we find that the maximum decay rate equals the maximum eigenvalue of the inverse covariance matrix, and that the infimum of the attainable multiplicative constant is 1, corresponding to the high-rotational limit in the Fokker-Planck drift. This analysis is complemented with numerical illustrations in 2D, and it includes a case study for time-dependent coefficient matrices.</p>